A more efficient approach for real materials consists in determining the “entanglement depth” [1]}, [2]}, [3]}, [4]}, defined as the minimum number of entangled modes required to construct a specific many-body state. The bound of entanglement depth can be quantified through expectation values of specific operators referred to as entanglement witnesses [5]}, [6]}, [7]}, [8]}, [9]}. Recent developments along this direction have been enabled by the use of the quantum Fisher information (QFI) [10]}, [11]}, [12]}, which can be quantified by measuring the equilibrium spectrum [13]}. By extracting the QFI from normalized material spectra, it is possible to witness a lower bound for the entanglement depth without relying on a full tomography of the wavefunction [14]}, [15]}, [16]}. This approach has been successfully applied to the study of magnetic materials[17]}, [18]}, [19]}, demonstrating its feasibility in solid-state experiments at equilibrium.
